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Chapter 3: Q7P (page 159)

Generalize Problem 6 to three dimensions; to n dimensions.

Short Answer

Expert verified

$C$ is an orthogonal matrix.

Step by step solution

01

Given information

C is a matrix whose columns are the components $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ of two perpendicular vectors, each of unit length.

02

Orthogonal matrix

An orthogonal matrix, also known as an orthonormal matrix, is a real square matrix with orthonormal vectors in the columns and rows.

03

Calculate matrix C

To generalize matrix $C$ , we need to represent n-dimensional matrix. Therefore, we will need $n$ orthogonal vectors $r_{n}$ each with $n$ components.

A n-dimensional matrix can be mathematically presented as

$C = (r_{1} r_{2} r_{2} . . . . . . . r_{n}), w h e r e r_{i}^{T} r_{j} = δ_{i j}, a n d δ_{i j} = \{\begin{matrix} 1 \\ 0 \end{matrix} \begin{matrix} i = j \\ i \neq j \end{matrix}$

Now, calculate $C^{T} C$ .

$C^{T} C = (\begin{matrix} r_{1}^{T} \\ r_{2}^{T} \\ ⋰ \\ . . \\ r_{n}^{T} \end{matrix}) (r_{1} r_{2} r_{2} . . . . . . . . . r_{n})$

Therefore, ${(C^{T} C)}_{i j} = r_{i}^{T} r_{j} = δ_{i j}$ , which shows that $C$ is an orthogonal matrix.

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Most popular questions from this chapter

In Problems 8 to 15,use to show that the given functions are linearly independent.

15. $e^{x}, e^{i x}, \cosh x$

Question: Find the values of $λ$ such that the following equations have nontrivial solutions, and for each $λ$ , solve the equations.

Question: Show that the unit matrix lhas the property that we associate with the number 1, that is,IA = AandAI = A, assuming that the matrices are conformable.

Show that a real Hermitian matrix is symmetric. Show that a real unitary matrix is orthogonal. Note: Thus, we see that Hermitian is the complex analogue of symmetric, and unitary is the complex analogue of orthogonal. (See Section 11.)

The Pauli spin matrices in quantum mechanics are $A = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$ , $B = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix})$ , $C = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$ .For the Pauli spin matrix C , find the matrices $sinkC$ , $coskC$ , $e^{kC}$ , and $e^{ikC}$ . Hint: Show that if a matrix is diagonal, say $D = (\begin{matrix} a & 0 \\ 0 & b \end{matrix})$ , then $f (D) = (\begin{matrix} f (a) & 0 \\ 0 & f (b) \end{matrix})$ .

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